Calculating the Perpendicular Height of a Trapezium Given Its Area and Parallel Sides

A trapezium (also known as a trapezoid in the U.S.) is a quadrilateral with exactly one pair of parallel sides. Understanding the properties and formulas associated with a trapezium can be valuable for various applications in geometry, engineering, and science. One common problem involves finding the height (or perpendicular distance) between the parallel sides when you know the area and the lengths of the parallel sides. In this article, we will explore how to solve such a problem and highlight the importance of consistent units in mathematical problem solving.

Problem: Calculating the Perpendicular Height of a Trapezium

Consider a trapezium with an area of 1500 mm2, and the lengths of its parallel sides are 4.8 cm and 5.2 cm. We need to determine the perpendicular height of the trapezium.

Step 1: Convert Units to Consistency

First, let's convert the lengths of the parallel sides from centimeters to millimeters, as the area is given in square millimeters.

Length of one parallel side:
4.8 cm 48 mm (since 1 cm 10 mm)

Length of the other parallel side:
5.2 cm 52 mm (since 1 cm 10 mm)

Step 2: Apply the Area Formula

The formula for the area of a trapezium is given by:

[text{Area} frac{1}{2} times (a b) times h]

Where:

a is the length of one of the parallel sides (48 mm)

b is the length of the other parallel side (52 mm)

h is the perpendicular height of the trapezium, which we are trying to find

Substituting the values we have:

[1500 frac{1}{2} times (48 52) times h]

Now, calculate the sum of the parallel sides:

[48 52 100]

Substituting back into the formula:

[1500 frac{1}{2} times 100 times h]

Simplifying further:

[1500 50h]

Step 3: Solve for the Height

Now, solve for h by dividing both sides by 50:

[h frac{1500}{50} 30 text{ mm}]

Therefore, the perpendicular height of the trapezium is 30 mm.

Additional Insights

Understanding the relationship between the area, parallel sides, and height of a trapezium is crucial in geometry. The trapezium problem we considered has practical applications in fields such as architecture, engineering, and design. It's also worth noting that the terms 'trapezium' and 'trapezoid' are defined differently in the UK and the US, which adds an interesting linguistic nuance to the subject.

To summarize, the key points we covered:

The formula for the area of a trapezium: (text{Area} frac{1}{2} times (a b) times h) Importance of consistent units when solving geometric problems A real example of how to apply the formula and solve for the height